If it overheats, the supply authority has badly f***ed up!
It might blow the service fuse perhaps.
Some flats do not have a gas supply,so rely on electric heating. Traditionally this has been via the dreaded night storage heaters.
Some of these flats have been retro-fitted with systems using traditional wet radiators. The boilers used are electrical, and use 3x4kW immersion heaters. When they were first being installed, the contractors were convinced they would overload the supply, and queried it with the manufacturer. They were reassured it would work; the control system constantly monitors supply voltage, and if it drops too low it starts switching off elements until none are on.
In practice it works well. That loading would be well above what is required for charging a Nissan Leaf, for example. With a smart charging system employing a similar strategy, local distribution won't be the huge problem you are imagining, although I accept that overall capacity in the country will need to improve.
Chris